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Positive solutions for Choquard equation in exterior domains

  • * Corresponding author

    * Corresponding author 
P. Chen was supported by the Research Foundation of Education Bureau of Hubei Province, China(Grant No.Q20192505). X. Liu was supported by the NSFC (Grant No.11771342)
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  • This work concerns with the following Choquard equation

    $ \begin{equation*} \begin{cases} -\Delta u+ u = (\int_{\Omega}\frac{u^2(y)}{|x-y|^{N-2}}dy)u &{\rm{in }}\; \Omega ,\\ u\in H_0^1(\Omega), \end{cases} \end{equation*} $

    where $ \Omega\subseteq \mathbb{R}^{N} $ is an exterior domain with smooth boundary. We prove that the equation has at least one positive solution by variational and toplogical methods. Moreover, we establish a nonlocal version of global compactness result in unbounded domain.

    Mathematics Subject Classification: Primary: 35J20, 35J60; Secondary: 35A16.


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